- Email: [email protected]
Investigation of hydrodynamic instability growth in copper
ARTICLE IN PRESS
JID: MS
[m5GeSdc;August 29, 2017;13:12]
International Journal of Mechanical Sciences 000 (2017) 1–6
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Investigation of hydrodynamic instability growth in copper Z. Sternberger a,∗, Y. Opachich c, C. Wehrenberg b, R. Kraus b, B. Remington b, N. Alexander d, G. Randall d, M. Farrell d, G. Ravichandran a a
California Institute of Technology, USA Lawrence Livermore National Laboratory, USA c National Security Technologies, LLC, USA d General Atomics, USA b
a r t i c l e
i n f o
Keywords: Hydrodynamic instability Copper Material strength
a b s t r a c t Hydrodynamic instability experiments allow access to material properties at strain rates above 105 s−1 . Existing models of material strength are not extensively validated at these strain rates. Recovery instability experiments were developed for copper samples. In conjunction with simulations using the hydrocode CTH, these experiments constrain the strength of copper at strain rates near 107 s−1 , providing an alternative technique to validate material properties at these strain rates. Mesh sensitivity in the present simulations contributed large uncertainty to the strength inferred from recovery instability experiments. © 2017 Published by Elsevier Ltd.
1. Introduction The strain rate dependence of the strength of copper has been extensively investigated for strain rates up to 104 s−1 using Hopkinson bar experiments [1,2]. Pressure-shear experiments have provided data up to strain rates of 106 s−1 [3–5]. Laser ablation coupled with dynamic diffraction has produced strength measurements at strain rates near 107 s−1 [6] and 1010 s−1 [7]. Strength inferred from overdriven shocks constrains the highest strain rates, above 109 s−1 [8]. The Preston–Tonks–Wallace (PTW) strength model bridges these many decades of strain rate behavior and has been calibrated to existing data for copper [9]. The model accounts for a thermal activation regime which transitions into a phonon drag regime as the strain rate increases. Data at strain rates where the regimes transition (from 105 to 109 s−1 ) are sparse, motivating new techniques to access this strain rate range. Hydrodynamic instability experiments have been developed to fill this gap. The Richtmyer–Meshkov instability (RMI) develops when a shock wave moves perpendicularly through a planar interface separating two fluids [10,11]. The interface is assumed to have an initial sinusoidal perturbation. If the amplitude of the perturbation is small with respect to the wavelength and compressibility is neglected, then the amplitude 𝜂(t) will grow with
where k is the perturbation wavenumber, 𝜂(0) is the initial perturbation amplitude, U is the interface velocity imparted by the shock, and 𝐴 = (𝜌2 − 𝜌1 )∕(𝜌2 + 𝜌1 ) is the Atwood number. The shock moves from material 1 with density 𝜌1 to material 2. If one of the materials is a solid, the strength of the solid will slow and ultimately arrest the growth of the instability. Rapid instability growth rate offers a window into the high strain rate behavior of the solid. If the shock transits to a low density material or a vacuum, then A < 0 and the amplitude inverts before growing. This configuration has been used to infer the strength of copper at strain rates near 107 s−1 [12–14]. 2. Experimental methods An initial perturbation is manufactured on the surface the experimental sample. The sample is assembled into a target consisting of an ablator, the sample, and a tamper. Laser ablation generates a blast wave which runs through the ablator and drives hydrodynamic instability at the ablator-sample interface (Fig. 1). The strength of the dynamically compressed sample arrests the instability growth and the tamper protects the sample against damage. The sample is recovered in a catcher [15] and growth of the initial perturbation amplitude due to hydrodynamic instability can be measured. 2.1. Sample recovery
𝜂(𝑡) = 𝜂(0)(1 + 𝑘𝑈 𝐴𝑡)
(1) Cylindrical copper samples with a 3 mm diameter were manufactured with an initial perturbation pattern and assembled into targets by
∗
Corresponding author. E-mail address: [email protected] (Z. Sternberger).
http://dx.doi.org/10.1016/j.ijmecsci.2017.08.051 Received 30 March 2017; Received in revised form 24 August 2017; Accepted 26 August 2017 Available online xxx 0020-7403/© 2017 Published by Elsevier Ltd.
Please cite this article as: Z. Sternberger et al., International Journal of Mechanical Sciences (2017), http://dx.doi.org/10.1016/j.ijmecsci.2017. 08.051
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 1. A pulsed laser drives a blast wave through the ablator, (a) and (b). The blast wave causes hydrodynamic instability at the perturbed ablator-sample interface, which is arrested after some growth. After the experiment, the sample is recovered (c).
Fig. 2. Images of cleaned recovered samples. The 2D initial perturbation runs horizontally in (a). The sample in (b) was manufactured with a 3D initial perturbation. The scratch on the left of the sample is a fiducial to help align pre- and post-shot profiles. The dark square is a remnant of the blast loading.
Fig. 3. The breakout velocity at the Cu-vacuum or Cu-LiF interface, depending on the drive target type, as a function of the laser energy used in the experiment, Es . The data points are the velocities measured by VISAR, for targets with and without a LiF window. The curves are a fit of breakout velocities determined from HYADES simulations, calibrated to the experimental points by choosing F.
General Atomics [16]. Two sinusoidal initial perturbation patterns were used, a 2D perturbation pattern and a 3D perturbation ( ) 𝜂(𝑥, 𝑦) = 𝜂0 sin 2𝜋𝑥∕𝜆𝑥
(2)
( ) ( ) 𝜂(𝑥, 𝑦) = 𝜂0 sin 2𝜋𝑥∕𝜆𝑥 sin 2𝜋𝑦∕𝜆𝑦
(3)
2.2. Analysis of laser ablation loading Drive targets, consisting of a 250 μm long CRF ablator and a 15 μm copper sample, were used to calibrate the energy in the laser pulse to the shocked state in the sample. One drive target included a lithium fluoride (LiF) window on the rear surface of the sample. During a drive shot, the velocity of the rear surface of the copper foil was measured with line VISAR [19,20]. Increasing the energy in the laser pulse increased the peak breakout velocity (Fig. 3). The relationship between breakout velocity and consequently peak pressure in the sample were interpolated using simulations in HYADES, a one dimensional radiation hydrodynamics code [21]. Simulated drive target experiments were matched to measured results using a scaling factor 𝐹 = 𝐸𝐻 ∕𝐸𝑠 , where Es is the laser energy used in the experiment (shot energy) and EH is the laser energy needed to replicate the measured breakout velocity in a HYADES simulation (HYADES energy). Two sets of simulations were run, with and without a LiF window on the drive target. A value of 0.74 was chosen for F so that a power law fit to simulation results best matched experimental results, both with and without the LiF window (Fig. 3). An uncertainty of ± 250 m/s was selected to encompass variations from the best fit line.
respectively, where 𝜆x and 𝜆y are the perturbation wavelengths and 𝜂 0 is the initial amplitude. For the 2D perturbation, 𝜆𝑥 = 50 𝜇m. For the 3D perturbation, 𝜆𝑥 = 𝜆𝑦 = 100 𝜇m. Both patterns used a 5 𝜇m initial amplitude. The manufactured initial perturbation was profiled with white light interferometry (Wyko NT9100). For a 3D perturbation where 𝜆𝑥 = 𝜆𝑦 , the linear instability growth develops according to the √ wavenumber 𝑘 = 2𝑘𝑥 [17,18]. Consequently, the linear growth rate of the 3D initial perturbation was slower than the growth rate of the 50 μm wavelength 2D initial perturbation used in experiment. Carbon resorcinol formaldehyde (CRF) foam with a density of approximately 0.115 g/cm3 was used as the ablator. The CRF was machined into cylinders 250 μm long and glued to the perturbed surface of the sample by tacking the edges of the ablator and the sample together. A thick tamper was tacked to the opposite face of the sample. During the experiment, the ablator was illuminated with a 3 ns pulse of 527 nm light using the Janus laser at the Jupiter Laser Facility, Lawrence Livermore National Laboratory. The laser spot was focused to a square roughly 1 mm on a side and the energy in the amplified pulse was varied to scan a range of peak pressures. After the experiment, the sample was cleaned of ablator debris (Fig. 2) and profiled again.
2.3. Measurement of instability growth The records of the sample profiles taken before and after the experiment were analyzed to determine the growth of the initial perturbations. Local extrema were identified in the shot region. The peak to valley am2
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 4. Extrema located on a post-shot 3D sample profile (a) and on a post-shot 2D sample profile (b). Blue circles locate maxima and red squares locate minima. White lines point to the neighbors of an example minima. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. SEM images of a 2D copper sample recovered from a peak pressure of 10 GPa. A region roughly the diameter of the laser spot is shown in (a). Two wavelengths of the perturbation near the shot center are shown in (b).
plitude (PTV), the average difference in height between an extrema and its neighboring extrema, were calculated (Fig. 4). The initial perturbations were deformed in a region surrounding the laser spot (Fig. 5). At the edges of the laser spot the loading was unconstrained and waves releasing the compressed ablator to ambient pressure propagated inward. The release could not cross the radius of the laser spot in the time the blast wave took to transit the ablator, leaving an undisturbed region at the shot center. The PTV in a 100 μm radius of the shot center, within this one dimensional loading region, were averaged to produce a single post-shot amplitude with an uncertainty of the standard deviation. This procedure was repeated using the record of the pre-shot profile. The peak pressure experienced by each recovered sample was determined from laser ablation simulations calibrated with VISAR data. As more energy was deposited in the ablator by the laser, the particle velocity and pressure induced at the ablator-sample interface by the blast wave increased. The increase in particle velocity lead to higher initial growth rates and a higher recovered growth factor, the ratio of the postshot amplitude to the pre-shot amplitude (Fig. 6). The larger wavelength 3D initial perturbations grew at a lower rate than the 2D initial perturbations at the same peak pressure. Consequently, the growth factor in samples manufactured with the 3D initial perturbation was lower at the same peak pressure than in samples with the 2D initial perturbation.
imental results were compared to simulations using the hydrocode CTH [22]. If the strength model used in simulations was incorrect, simulated growth would be different than growth measured in experiment when subjected to the same loading condition. Simulations used an ideal gas equation of state for the CRF ablator with 𝛾 = 1.3 so that, in the strong shock limit, the slope of the Hugoniot matched the experimentally determined Hugoniot of carbon aerogel at a similar density of 0.12 g/cc [23]. The Sesame equation of state [24] and the Steinberg–Guinan strength model (SG) [25] were used for the copper sample. The Steinberg–Guinan strength model predicts that the material expresses pressure and work hardening, but assumes the strain rate dependence has saturated at high strain rates. In the PTW model, the strength continues to increase with strain rate. The Steinberg–Guinan strength model can be scaled as an approximation of strain rate hardening. For example, the Steinberg–Guinan strength model was scaled by 2 by increasing the ambient yield stress and the maximum allowed yield stress by a factor of 2. For copper, the ambient yield stress was 120 MPa and the maximum allowed yield stress due to work hardening was 640 MPa [26]. CTH cannot simulate laser ablation. Simulations in HYADES determined that depositing energy in the first 1 μm of the ablator generated comparable blast waves to those generated in laser ablation simulations. In 1D CTH simulations, energy was deposited in the first 1 μm of the ablator at initial time, causing this material to expand and launch a blast wave through the ablator. The state of the ablator after the blast wave had traveled within 50 μm of the ablator-sample interface was saved and used to initialize subsequent simulations of the interaction of the blast wave with the sample.
3. Simulation of instability growth Because the sample was not loaded by a supported shock, the initial perturbation did not undergo classic RMI and the growth factor could not be compared to models of RMI growth in solids. Instead, the exper3
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 6. Growth factors determined from recovered samples. Plots are for 2D (a) and 3D (b) initial perturbations.
Fig. 7. Convergence of peak pressure with zone width 𝓁 for 1D simulations (a) and convergence of the final growth factor with 𝓁 for simulations of samples with 2D (b) and 3D (c) initial perturbations. The 2D convergence rate was assumed to hold for 3D simulations.
mesh convergence was fit well with a power law based on 𝓁, the zone length. The fit predicted the final growth factor would slightly increase beyond the highest mesh resolution attained in simulations. The error between the limit of the power law and the final growth factor obtained with the 0.17 μm zone length was 11.6%. The 3D simulations were too computationally expensive to resolve much of the convergence behavior. It was assumed that the rate of the convergence of the 2D and 3D simulations were similar, so the exponent of the power law fit to 3D mesh convergence was set from the 2D power law fit. The error between the limit of the fit and the value obtained with the 0.83 μm zone length was 16%. A number of 2D and 3D simulations were run, covering a range of energy deposited in the ablator and strength model scalings from 1/4 to 10. The state of the sample in 2D or 3D simulations was difficult to describe because the shock wave distorts when passing through the interface perturbation. For each 2D or 3D simulation, a corresponding 1D simulation was run using the same loading condition and strength
A mesh of constant length zones with aspect ratio 1.0 was used to simulate the ablator and 50 μm of the sample. The mesh was extended at lower resolution beyond the sample to prevent any reflections from the edge of the meshed domain from interfering with the instability growth at the ablator-sample interface. One half wavelength of the 2D initial perturbation, from peak to valley, was simulated. The resolution of the mesh was 0.17 μm per zone. Simulations of 3D initial perturbations required a 3D domain, covering one eight of the initial perturbation area in order to capture one peak and one valley. The more computationally expensive 3D simulations were run with a coarser resolution than the 2D simulations at 0.83 μm per zone. Uncertainty due to mesh sensitivity was estimated with a mesh convergence study using the unmodified SG strength model and a blast wave loading which produced a peak pressure in the copper of roughly 15 GPa. In the 2D mesh convergence study, the final growth factor increased with increasing mesh resolution, approaching a final value (Fig. 7). The 4
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
The experimental results validate a range of simulations that produce the same growth factor at the same peak pressure. However, the mesh sensitivity of the simulations lends some uncertainty to the comparison, equivalent to shifting the simulations to higher growth factor and higher peak pressure. To account for this uncertainty, the experimental errorbars were extended to lower growth factor and lower peak pressure. The smallest and largest growth strengths encompassed by each experimental point were taken as the range of strengths of the sample material at that strain rate. Some of the lower growth factor results from the 3D initial perturbation samples shown in Fig. 6 were excluded as the adjusted errorbars extended to growth factors lower than 1.0 and no simulations were available for comparison. 5. Results and conclusions Fig. 8. The growth factor in a 2D simulation using the Steinberg–Guinan strength model scaled by 2 compared to the pressure at the ablator-sample interface in the corresponding 1D simulation. The pressure released from its peak value as the instability growth started.
Each comparison between experiment and simulation produced one inferred growth strength and corresponding strain rate. Within uncertainty, all of the inferred strain rates covered the same 107 s−1 strain rate range. Uncertainty in each inferred strength was large, extending 2 to 4 GPa over the mean inferred growth strength of approximately 700 MPa. In order to bring down the uncertainty, all the inferred strengths were averaged for each initial perturbation (Fig. 10). The inferred strengths sit in the transition between the thermal activation regime and the phonon drag regime. The strength inferred from the mean growth factor for 2D initial perturbations, 730 MPa at a strain rate of roughly 2 × 107 s−1 , was slightly less than the approximately 750 to 800 MPa strength predicted by the PTW model. However, even after the averaging, the uncertainty encompassed strengths 1 to 2 GPa larger than the mean inferred strength. Due to variation of the PTV in the shot center, the uncertainty in the growth factor was roughly ± 0.5. In simulations at a given peak pressure, slightly lowering the strength on the order of 100 MPa could account for the +0.5 uncertainty. Large increase in strength on the order of 1 GPa were necessary to account for the −0.5 uncertainty. Recent work inferring the strength of copper from RMI experiments with A < 0 found a strength of 530 ± 100 MPa at strain rates between 2 × 106 s−1 and 1.6 × 107 s−1 [14]. For these results, the initial perturbation was imposed on the free surface of the sample so that the instability growth rate could be measured with laser velocimetry (PDV). The measured growth rate was then compared to simulations of the instability growth. Because the initial perturbation was on the free surface of the sample, the sample was unconstrained and released quickly to low pressure during instability growth. Consequently, the state of the samples in the A < 0 experiments and in the recovery experiments with A > 0 were not identical despite a similar strain rate. The uncertainty in the strength inferred from recovery instability experiments could be reduced two ways. First, the uncertainty due to mesh sensitivity could be improved with sufficiently resolved simulations. It is expected that fully converged simulations would approximately half the uncertainty of the inferred strength. Second, the uncertainty in sample state could have been improved with additional VISAR measurements. However, the peak pressure generated in experiment was sensitive to the laser energy deposited on the CRF ablator. Uncertainty in specifying the delivered laser energy was magnified by this high sensitivity, which made it difficult to take VISAR measurements in the desired range of peak pressures. In conclusion, we have recovered copper samples which underwent deformation due to hydrodynamic instability. The measured growth factors are useful validation points for hydrocode simulations. We simulated the instability growth in CTH, covering a range of growth factors and peak pressures seen in experiment. The uncertainty in the inferred strength was large, but could be reduced by improved simulations of instability growth. Future experiments could further reduce uncertainty by securing more VISAR data to better understand the sample state and by using an ablator which is less sensitive to the deposited laser energy.
model scaling. In 1D simulations, the peak pressure in the copper increased as the resolution of the simulation was increased, converging to a final value. A mesh resolution of 0.17 μm zone length was sufficient to produce the converged value of peak pressure within 1% and was used for all 1D simulations. 4. Comparing simulations of instability growth and experimental results The growth factor and peak pressure measured in both simulation and experiments were compared. The experimental peak pressure was determined from HYADES simulations calibrated to VISAR measurements and the growth factor was determined from the profiles of the sample pre- and post-shot. The simulation results are described by the energy used to generate the blast wave loading and the strength model scaling. The simulated peak pressure was determined from a 1D simulation of the loading, with uncertainty due to mesh sensitivity. The simulated growth factor was determined from a 2D or 3D simulation of the loading, with uncertainty due to mesh sensitivity. In order to calculate the strength of the sample, the states from both the 1D simulation and information about the strain and strain rate from the 2D or 3D simulation were required. The state of the sample was time-varying during the instability growth (Fig. 8). The time when the perturbation amplitude was growing was identified in 2D or 3D simulations and the state of the sample near the ablator-sample interface in the corresponding 1D simulation was averaged during this time. This average state during growth, or growth state, is a simple description of the density, pressure, and temperature experienced by the sample. The strain in the sample was approximated by calculating the strain of the amplitude during the instability growth, starting from the amplitude immediately after the perturbation was compressed by the shock transit. The strain rate was approximated with a linear fit to the strain in time. Together with the growth pressure, density, and temperature, the maximum strain was input into the scaled Steinberg–Guinan strength model to calculate the growth strength. The growth strength is only an average of the strength of the sample; the strength peaked at the beginning of instability growth when the pressure was its largest and then decreased as the sample released. The growth states and the strain rate were linearly interpolated between the simulations. The growth strength, calculated from the scaled Steinberg–Guinan strength model, was interpolated between the simulations as well (Fig. 9). As the peak pressure in simulations increased, melt in the perturbation valley allowed some sample material to flow away from the ablator-sample interface and complicated calculating a growth factor. Consequently, some of the higher peak pressure results from Fig. 6 were excluded because no simulations were available for comparison. 5
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 9. An overlay of simulation results (evaluated at the gray points) and experimental results (black points). The growth strength calculated from each simulation was interpolated over the range of peak pressures and growth factors seen in experiment for samples with 2D (a) and 3D (b) initial perturbations. Curves of constant growth strength are in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
[5] Tong W. Pressure-shear stress wave analysis in plate impact experiments. Int J Impact Eng 1997;19(2):147–64. [6] Meyers MA, Gregori F, Kad BK, Schneider MS, Kalantar DH, Remington BA, et al. Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Materialia 2003;51(5):1211–28. doi:10.1016/S1359-6454(02)00420-2. [7] Murphy WJ, Higginbotham A, Kimminau G, Barbrel B, Bringa EM, Hawreliak J, et al. The strength of single crystal copper under uniaxial shock compression at 100 GPa. J Phys 2010;22(6):065404. doi:10.1088/0953-8984/22/6/065404. [8] Wallace DC. Nature of the process of overdriven shocks in metals. Phys Rev B 1981;24(10):5607–15. doi:10.1103/PhysRevB.24.5607. [9] Preston DL, Tonks DL, Wallace DC. Model of plastic deformation for extreme loading conditions. J Appl Phys 2003;93(1):211–20. doi:10.1063/1.1524706. [10] Richtmyer RD. Taylor instability in shock acceleration of compressible fluids. Commun Pure Appl Math 1960;13(2):297–319. doi:10.1002/cpa.3160130207. [11] Meshkov EE. Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn 1969;4(5):101–4. doi:10.1007/BF01015969. [12] Prime MB, Vaughan DE, Preston DL, Buttler WT, Chen SR, Oró DM, et al. Using growth and arrest of richtmyer-meshkov instabilities and lagrangian simulations to study high-rate material strength. J Phys 2014;500(11):112051. doi:10.1088/1742-6596/500/11/112051. [13] Prime MB, Buttler WT, Sjue SK, Jensen BJ, Mariam FG, Oró DM, et al. Using Richtmyer-Meshkov instabilities to estimate metal strength at very high rates. In: Song B, Lamberson L, Casem D, Kimberley J, editors. Dynamic Behavior of Materials, Volume 1. Springer International Publishing; 2016. p. 191–7. ISBN:978-3-31922451-0 978-3-319-22452-7, doi: 10.1007/978-3-319-22452-7_27. [14] Prime MB, Buttler WT, Buechler MA, Denissen NA, Kenamond MA, Mariam FG, et al. Estimation of metal strength at very high rates using free-surface richtmyermeshkov instabilities. J Dyn Behav Mater 2017:1–14. doi:10.1007/s40870-017-0103-9. [15] Maddox BR, Park HS, Lu CH, Remington BA, Prisbrey S, Kad B, et al. Isentropic/shock compression and recovery methodology for materials using high-amplitude laser pulses. Mater Sci Eng 2013;578:354–61. doi:10.1016/j.msea.2013.04.050. [16] Randall GC, Vecchio J, Knipping J, Wall D, Remington T, Fitzsimmons P, et al. Developments in microcoining rippled metal foils. Fus Sci Technol 2013;63(2):274–81 URL http://epubs.ans.org/?a=16350. [17] Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Courier Corporation; 2013. [18] Jacobs JW, Catton I. Three-dimensional rayleigh-taylor instability part 2. experiment. J Fluid Mech 1988;187:353–71. doi:10.1017/S0022112088000461. [19] Barker LM, Hollenbach RE. Laser interferometer for measuring high velocities of any reflecting surface. J Appl Phys 1972;43(11):4669–75. doi:10.1063/1.1660986. [20] Celliers PM, Bradley DK, Collins GW, Hicks DG, Boehly TR, Armstrong WJ. Lineimaging velocimeter for shock diagnostics at the OMEGA laser facility. Rev Sci Instrum 2004;75(11):4916–29. doi:10.1063/1.1807008. [21] Larsen JT, Lane SM. HYADES a plasma hydrodynamics code for dense plasma studies. J Quant Spectrosc Radiat Transf 1994;51(12):179–86. doi:10.1016/0022-4073(94)90078-7. [22] McGlaun JM, Thompson SL, Elrick MG. CTH: A three-dimensional shock wave physics code. Int J Impact Eng 1990;10(1):351–60. doi:10.1016/0734-743X(90)90071-3. [23] Holmes NC. Shock compression of low-density foams. In: AIP Conference Proceedings, 309. AIP Publishing; 1994. p. 153–6. doi:10.1063/1.46377. [24] Bennett BI, Johnson JD, Kerley GI, Rood GT. Recent developments in the sesame equation-of-state library. Tech. Rep., LA-7130. Los Alamos Scientific Lab., N.Mex. (USA); 1978. URL http://www.osti.gov/scitech/biblio/5150206. [25] Steinberg DJ, Cochran SG, Guinan MW. A constitutive model for metals applicable at high-strain rate. J Appl Phys 1980;51(3):1498–504. doi:10.1063/1.327799. [26] Steinberg DJ. Equation of state and strength properties of selected materials. Tech. Rep., UCRL-MA-106439 (change 1). Lawrence Livermore National Laboratory, CA (USA); 1996.
Fig. 10. A comparison between the strength inferred from recovered samples with 2D and 3D initial perturbations and the PTW model. Two PTW curves using different states were calculated for comparison. The first curve used 𝑃 = 0 GPa, 𝑇 = 300 K, and plastic strain 𝜀 = 0.25 as a lower bound of growth states. The second curved is an upper bound with 𝑃 = 5 GPa, 𝑇 = 600 K, and 𝜀 = 1.5.
Acknowledgments This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Security, LLC, Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This work was supported by the NNSA through the HEDLP program, grant numbers DE-NA0001805 and DE-NA0001832. Experimental time was awarded by the Jupiter Laser Facility at Lawrence Livermore National Laboratory. We would like to thank the staff at the Jupiter Laser Facility for their help during our experimental campaign. The experiment was completed with the help of Professor Marc Meyers, Shiteng Zhao, and Eric Hahn of the University of California, San Diego; Suzanne Ali of the University of California, Berkeley; and Laura Chen of Imperial College, London. References [1] Kumar A, Kumble RG. Viscous drag on dislocations at high strain rates in copper. J Appl Phys 1969;40(9):3475–80. doi:10.1063/1.1658222. [2] Follansbee PS, Gray GT. Dynamic deformation of shock prestrained copper. Mater Sci Eng 1991;138(1):23–31. doi:10.1016/0921-5093(91)90673-B. [3] Frutschy KJ, Clifton RJ. High-temperature pressure-shear plate impact experiments on OFHC copper. J Mech Phys Solids 1998;46(10):1723–44. doi:10.1016/S0022-5096(98)00055-6. [4] Clifton RJ. Response of materials under dynamic loading. Int J Solids Struct 2000;37(12):105–13. doi:10.1016/S0020-7683(99)00082-7.
6
JID: MS
[m5GeSdc;August 29, 2017;13:12]
International Journal of Mechanical Sciences 000 (2017) 1–6
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Investigation of hydrodynamic instability growth in copper Z. Sternberger a,∗, Y. Opachich c, C. Wehrenberg b, R. Kraus b, B. Remington b, N. Alexander d, G. Randall d, M. Farrell d, G. Ravichandran a a
California Institute of Technology, USA Lawrence Livermore National Laboratory, USA c National Security Technologies, LLC, USA d General Atomics, USA b
a r t i c l e
i n f o
Keywords: Hydrodynamic instability Copper Material strength
a b s t r a c t Hydrodynamic instability experiments allow access to material properties at strain rates above 105 s−1 . Existing models of material strength are not extensively validated at these strain rates. Recovery instability experiments were developed for copper samples. In conjunction with simulations using the hydrocode CTH, these experiments constrain the strength of copper at strain rates near 107 s−1 , providing an alternative technique to validate material properties at these strain rates. Mesh sensitivity in the present simulations contributed large uncertainty to the strength inferred from recovery instability experiments. © 2017 Published by Elsevier Ltd.
1. Introduction The strain rate dependence of the strength of copper has been extensively investigated for strain rates up to 104 s−1 using Hopkinson bar experiments [1,2]. Pressure-shear experiments have provided data up to strain rates of 106 s−1 [3–5]. Laser ablation coupled with dynamic diffraction has produced strength measurements at strain rates near 107 s−1 [6] and 1010 s−1 [7]. Strength inferred from overdriven shocks constrains the highest strain rates, above 109 s−1 [8]. The Preston–Tonks–Wallace (PTW) strength model bridges these many decades of strain rate behavior and has been calibrated to existing data for copper [9]. The model accounts for a thermal activation regime which transitions into a phonon drag regime as the strain rate increases. Data at strain rates where the regimes transition (from 105 to 109 s−1 ) are sparse, motivating new techniques to access this strain rate range. Hydrodynamic instability experiments have been developed to fill this gap. The Richtmyer–Meshkov instability (RMI) develops when a shock wave moves perpendicularly through a planar interface separating two fluids [10,11]. The interface is assumed to have an initial sinusoidal perturbation. If the amplitude of the perturbation is small with respect to the wavelength and compressibility is neglected, then the amplitude 𝜂(t) will grow with
where k is the perturbation wavenumber, 𝜂(0) is the initial perturbation amplitude, U is the interface velocity imparted by the shock, and 𝐴 = (𝜌2 − 𝜌1 )∕(𝜌2 + 𝜌1 ) is the Atwood number. The shock moves from material 1 with density 𝜌1 to material 2. If one of the materials is a solid, the strength of the solid will slow and ultimately arrest the growth of the instability. Rapid instability growth rate offers a window into the high strain rate behavior of the solid. If the shock transits to a low density material or a vacuum, then A < 0 and the amplitude inverts before growing. This configuration has been used to infer the strength of copper at strain rates near 107 s−1 [12–14]. 2. Experimental methods An initial perturbation is manufactured on the surface the experimental sample. The sample is assembled into a target consisting of an ablator, the sample, and a tamper. Laser ablation generates a blast wave which runs through the ablator and drives hydrodynamic instability at the ablator-sample interface (Fig. 1). The strength of the dynamically compressed sample arrests the instability growth and the tamper protects the sample against damage. The sample is recovered in a catcher [15] and growth of the initial perturbation amplitude due to hydrodynamic instability can be measured. 2.1. Sample recovery
𝜂(𝑡) = 𝜂(0)(1 + 𝑘𝑈 𝐴𝑡)
(1) Cylindrical copper samples with a 3 mm diameter were manufactured with an initial perturbation pattern and assembled into targets by
∗
Corresponding author. E-mail address: [email protected] (Z. Sternberger).
http://dx.doi.org/10.1016/j.ijmecsci.2017.08.051 Received 30 March 2017; Received in revised form 24 August 2017; Accepted 26 August 2017 Available online xxx 0020-7403/© 2017 Published by Elsevier Ltd.
Please cite this article as: Z. Sternberger et al., International Journal of Mechanical Sciences (2017), http://dx.doi.org/10.1016/j.ijmecsci.2017. 08.051
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 1. A pulsed laser drives a blast wave through the ablator, (a) and (b). The blast wave causes hydrodynamic instability at the perturbed ablator-sample interface, which is arrested after some growth. After the experiment, the sample is recovered (c).
Fig. 2. Images of cleaned recovered samples. The 2D initial perturbation runs horizontally in (a). The sample in (b) was manufactured with a 3D initial perturbation. The scratch on the left of the sample is a fiducial to help align pre- and post-shot profiles. The dark square is a remnant of the blast loading.
Fig. 3. The breakout velocity at the Cu-vacuum or Cu-LiF interface, depending on the drive target type, as a function of the laser energy used in the experiment, Es . The data points are the velocities measured by VISAR, for targets with and without a LiF window. The curves are a fit of breakout velocities determined from HYADES simulations, calibrated to the experimental points by choosing F.
General Atomics [16]. Two sinusoidal initial perturbation patterns were used, a 2D perturbation pattern and a 3D perturbation ( ) 𝜂(𝑥, 𝑦) = 𝜂0 sin 2𝜋𝑥∕𝜆𝑥
(2)
( ) ( ) 𝜂(𝑥, 𝑦) = 𝜂0 sin 2𝜋𝑥∕𝜆𝑥 sin 2𝜋𝑦∕𝜆𝑦
(3)
2.2. Analysis of laser ablation loading Drive targets, consisting of a 250 μm long CRF ablator and a 15 μm copper sample, were used to calibrate the energy in the laser pulse to the shocked state in the sample. One drive target included a lithium fluoride (LiF) window on the rear surface of the sample. During a drive shot, the velocity of the rear surface of the copper foil was measured with line VISAR [19,20]. Increasing the energy in the laser pulse increased the peak breakout velocity (Fig. 3). The relationship between breakout velocity and consequently peak pressure in the sample were interpolated using simulations in HYADES, a one dimensional radiation hydrodynamics code [21]. Simulated drive target experiments were matched to measured results using a scaling factor 𝐹 = 𝐸𝐻 ∕𝐸𝑠 , where Es is the laser energy used in the experiment (shot energy) and EH is the laser energy needed to replicate the measured breakout velocity in a HYADES simulation (HYADES energy). Two sets of simulations were run, with and without a LiF window on the drive target. A value of 0.74 was chosen for F so that a power law fit to simulation results best matched experimental results, both with and without the LiF window (Fig. 3). An uncertainty of ± 250 m/s was selected to encompass variations from the best fit line.
respectively, where 𝜆x and 𝜆y are the perturbation wavelengths and 𝜂 0 is the initial amplitude. For the 2D perturbation, 𝜆𝑥 = 50 𝜇m. For the 3D perturbation, 𝜆𝑥 = 𝜆𝑦 = 100 𝜇m. Both patterns used a 5 𝜇m initial amplitude. The manufactured initial perturbation was profiled with white light interferometry (Wyko NT9100). For a 3D perturbation where 𝜆𝑥 = 𝜆𝑦 , the linear instability growth develops according to the √ wavenumber 𝑘 = 2𝑘𝑥 [17,18]. Consequently, the linear growth rate of the 3D initial perturbation was slower than the growth rate of the 50 μm wavelength 2D initial perturbation used in experiment. Carbon resorcinol formaldehyde (CRF) foam with a density of approximately 0.115 g/cm3 was used as the ablator. The CRF was machined into cylinders 250 μm long and glued to the perturbed surface of the sample by tacking the edges of the ablator and the sample together. A thick tamper was tacked to the opposite face of the sample. During the experiment, the ablator was illuminated with a 3 ns pulse of 527 nm light using the Janus laser at the Jupiter Laser Facility, Lawrence Livermore National Laboratory. The laser spot was focused to a square roughly 1 mm on a side and the energy in the amplified pulse was varied to scan a range of peak pressures. After the experiment, the sample was cleaned of ablator debris (Fig. 2) and profiled again.
2.3. Measurement of instability growth The records of the sample profiles taken before and after the experiment were analyzed to determine the growth of the initial perturbations. Local extrema were identified in the shot region. The peak to valley am2
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 4. Extrema located on a post-shot 3D sample profile (a) and on a post-shot 2D sample profile (b). Blue circles locate maxima and red squares locate minima. White lines point to the neighbors of an example minima. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. SEM images of a 2D copper sample recovered from a peak pressure of 10 GPa. A region roughly the diameter of the laser spot is shown in (a). Two wavelengths of the perturbation near the shot center are shown in (b).
plitude (PTV), the average difference in height between an extrema and its neighboring extrema, were calculated (Fig. 4). The initial perturbations were deformed in a region surrounding the laser spot (Fig. 5). At the edges of the laser spot the loading was unconstrained and waves releasing the compressed ablator to ambient pressure propagated inward. The release could not cross the radius of the laser spot in the time the blast wave took to transit the ablator, leaving an undisturbed region at the shot center. The PTV in a 100 μm radius of the shot center, within this one dimensional loading region, were averaged to produce a single post-shot amplitude with an uncertainty of the standard deviation. This procedure was repeated using the record of the pre-shot profile. The peak pressure experienced by each recovered sample was determined from laser ablation simulations calibrated with VISAR data. As more energy was deposited in the ablator by the laser, the particle velocity and pressure induced at the ablator-sample interface by the blast wave increased. The increase in particle velocity lead to higher initial growth rates and a higher recovered growth factor, the ratio of the postshot amplitude to the pre-shot amplitude (Fig. 6). The larger wavelength 3D initial perturbations grew at a lower rate than the 2D initial perturbations at the same peak pressure. Consequently, the growth factor in samples manufactured with the 3D initial perturbation was lower at the same peak pressure than in samples with the 2D initial perturbation.
imental results were compared to simulations using the hydrocode CTH [22]. If the strength model used in simulations was incorrect, simulated growth would be different than growth measured in experiment when subjected to the same loading condition. Simulations used an ideal gas equation of state for the CRF ablator with 𝛾 = 1.3 so that, in the strong shock limit, the slope of the Hugoniot matched the experimentally determined Hugoniot of carbon aerogel at a similar density of 0.12 g/cc [23]. The Sesame equation of state [24] and the Steinberg–Guinan strength model (SG) [25] were used for the copper sample. The Steinberg–Guinan strength model predicts that the material expresses pressure and work hardening, but assumes the strain rate dependence has saturated at high strain rates. In the PTW model, the strength continues to increase with strain rate. The Steinberg–Guinan strength model can be scaled as an approximation of strain rate hardening. For example, the Steinberg–Guinan strength model was scaled by 2 by increasing the ambient yield stress and the maximum allowed yield stress by a factor of 2. For copper, the ambient yield stress was 120 MPa and the maximum allowed yield stress due to work hardening was 640 MPa [26]. CTH cannot simulate laser ablation. Simulations in HYADES determined that depositing energy in the first 1 μm of the ablator generated comparable blast waves to those generated in laser ablation simulations. In 1D CTH simulations, energy was deposited in the first 1 μm of the ablator at initial time, causing this material to expand and launch a blast wave through the ablator. The state of the ablator after the blast wave had traveled within 50 μm of the ablator-sample interface was saved and used to initialize subsequent simulations of the interaction of the blast wave with the sample.
3. Simulation of instability growth Because the sample was not loaded by a supported shock, the initial perturbation did not undergo classic RMI and the growth factor could not be compared to models of RMI growth in solids. Instead, the exper3
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 6. Growth factors determined from recovered samples. Plots are for 2D (a) and 3D (b) initial perturbations.
Fig. 7. Convergence of peak pressure with zone width 𝓁 for 1D simulations (a) and convergence of the final growth factor with 𝓁 for simulations of samples with 2D (b) and 3D (c) initial perturbations. The 2D convergence rate was assumed to hold for 3D simulations.
mesh convergence was fit well with a power law based on 𝓁, the zone length. The fit predicted the final growth factor would slightly increase beyond the highest mesh resolution attained in simulations. The error between the limit of the power law and the final growth factor obtained with the 0.17 μm zone length was 11.6%. The 3D simulations were too computationally expensive to resolve much of the convergence behavior. It was assumed that the rate of the convergence of the 2D and 3D simulations were similar, so the exponent of the power law fit to 3D mesh convergence was set from the 2D power law fit. The error between the limit of the fit and the value obtained with the 0.83 μm zone length was 16%. A number of 2D and 3D simulations were run, covering a range of energy deposited in the ablator and strength model scalings from 1/4 to 10. The state of the sample in 2D or 3D simulations was difficult to describe because the shock wave distorts when passing through the interface perturbation. For each 2D or 3D simulation, a corresponding 1D simulation was run using the same loading condition and strength
A mesh of constant length zones with aspect ratio 1.0 was used to simulate the ablator and 50 μm of the sample. The mesh was extended at lower resolution beyond the sample to prevent any reflections from the edge of the meshed domain from interfering with the instability growth at the ablator-sample interface. One half wavelength of the 2D initial perturbation, from peak to valley, was simulated. The resolution of the mesh was 0.17 μm per zone. Simulations of 3D initial perturbations required a 3D domain, covering one eight of the initial perturbation area in order to capture one peak and one valley. The more computationally expensive 3D simulations were run with a coarser resolution than the 2D simulations at 0.83 μm per zone. Uncertainty due to mesh sensitivity was estimated with a mesh convergence study using the unmodified SG strength model and a blast wave loading which produced a peak pressure in the copper of roughly 15 GPa. In the 2D mesh convergence study, the final growth factor increased with increasing mesh resolution, approaching a final value (Fig. 7). The 4
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
The experimental results validate a range of simulations that produce the same growth factor at the same peak pressure. However, the mesh sensitivity of the simulations lends some uncertainty to the comparison, equivalent to shifting the simulations to higher growth factor and higher peak pressure. To account for this uncertainty, the experimental errorbars were extended to lower growth factor and lower peak pressure. The smallest and largest growth strengths encompassed by each experimental point were taken as the range of strengths of the sample material at that strain rate. Some of the lower growth factor results from the 3D initial perturbation samples shown in Fig. 6 were excluded as the adjusted errorbars extended to growth factors lower than 1.0 and no simulations were available for comparison. 5. Results and conclusions Fig. 8. The growth factor in a 2D simulation using the Steinberg–Guinan strength model scaled by 2 compared to the pressure at the ablator-sample interface in the corresponding 1D simulation. The pressure released from its peak value as the instability growth started.
Each comparison between experiment and simulation produced one inferred growth strength and corresponding strain rate. Within uncertainty, all of the inferred strain rates covered the same 107 s−1 strain rate range. Uncertainty in each inferred strength was large, extending 2 to 4 GPa over the mean inferred growth strength of approximately 700 MPa. In order to bring down the uncertainty, all the inferred strengths were averaged for each initial perturbation (Fig. 10). The inferred strengths sit in the transition between the thermal activation regime and the phonon drag regime. The strength inferred from the mean growth factor for 2D initial perturbations, 730 MPa at a strain rate of roughly 2 × 107 s−1 , was slightly less than the approximately 750 to 800 MPa strength predicted by the PTW model. However, even after the averaging, the uncertainty encompassed strengths 1 to 2 GPa larger than the mean inferred strength. Due to variation of the PTV in the shot center, the uncertainty in the growth factor was roughly ± 0.5. In simulations at a given peak pressure, slightly lowering the strength on the order of 100 MPa could account for the +0.5 uncertainty. Large increase in strength on the order of 1 GPa were necessary to account for the −0.5 uncertainty. Recent work inferring the strength of copper from RMI experiments with A < 0 found a strength of 530 ± 100 MPa at strain rates between 2 × 106 s−1 and 1.6 × 107 s−1 [14]. For these results, the initial perturbation was imposed on the free surface of the sample so that the instability growth rate could be measured with laser velocimetry (PDV). The measured growth rate was then compared to simulations of the instability growth. Because the initial perturbation was on the free surface of the sample, the sample was unconstrained and released quickly to low pressure during instability growth. Consequently, the state of the samples in the A < 0 experiments and in the recovery experiments with A > 0 were not identical despite a similar strain rate. The uncertainty in the strength inferred from recovery instability experiments could be reduced two ways. First, the uncertainty due to mesh sensitivity could be improved with sufficiently resolved simulations. It is expected that fully converged simulations would approximately half the uncertainty of the inferred strength. Second, the uncertainty in sample state could have been improved with additional VISAR measurements. However, the peak pressure generated in experiment was sensitive to the laser energy deposited on the CRF ablator. Uncertainty in specifying the delivered laser energy was magnified by this high sensitivity, which made it difficult to take VISAR measurements in the desired range of peak pressures. In conclusion, we have recovered copper samples which underwent deformation due to hydrodynamic instability. The measured growth factors are useful validation points for hydrocode simulations. We simulated the instability growth in CTH, covering a range of growth factors and peak pressures seen in experiment. The uncertainty in the inferred strength was large, but could be reduced by improved simulations of instability growth. Future experiments could further reduce uncertainty by securing more VISAR data to better understand the sample state and by using an ablator which is less sensitive to the deposited laser energy.
model scaling. In 1D simulations, the peak pressure in the copper increased as the resolution of the simulation was increased, converging to a final value. A mesh resolution of 0.17 μm zone length was sufficient to produce the converged value of peak pressure within 1% and was used for all 1D simulations. 4. Comparing simulations of instability growth and experimental results The growth factor and peak pressure measured in both simulation and experiments were compared. The experimental peak pressure was determined from HYADES simulations calibrated to VISAR measurements and the growth factor was determined from the profiles of the sample pre- and post-shot. The simulation results are described by the energy used to generate the blast wave loading and the strength model scaling. The simulated peak pressure was determined from a 1D simulation of the loading, with uncertainty due to mesh sensitivity. The simulated growth factor was determined from a 2D or 3D simulation of the loading, with uncertainty due to mesh sensitivity. In order to calculate the strength of the sample, the states from both the 1D simulation and information about the strain and strain rate from the 2D or 3D simulation were required. The state of the sample was time-varying during the instability growth (Fig. 8). The time when the perturbation amplitude was growing was identified in 2D or 3D simulations and the state of the sample near the ablator-sample interface in the corresponding 1D simulation was averaged during this time. This average state during growth, or growth state, is a simple description of the density, pressure, and temperature experienced by the sample. The strain in the sample was approximated by calculating the strain of the amplitude during the instability growth, starting from the amplitude immediately after the perturbation was compressed by the shock transit. The strain rate was approximated with a linear fit to the strain in time. Together with the growth pressure, density, and temperature, the maximum strain was input into the scaled Steinberg–Guinan strength model to calculate the growth strength. The growth strength is only an average of the strength of the sample; the strength peaked at the beginning of instability growth when the pressure was its largest and then decreased as the sample released. The growth states and the strain rate were linearly interpolated between the simulations. The growth strength, calculated from the scaled Steinberg–Guinan strength model, was interpolated between the simulations as well (Fig. 9). As the peak pressure in simulations increased, melt in the perturbation valley allowed some sample material to flow away from the ablator-sample interface and complicated calculating a growth factor. Consequently, some of the higher peak pressure results from Fig. 6 were excluded because no simulations were available for comparison. 5
JID: MS
ARTICLE IN PRESS
Z. Sternberger et al.
[m5GeSdc;August 29, 2017;13:12] International Journal of Mechanical Sciences 000 (2017) 1–6
Fig. 9. An overlay of simulation results (evaluated at the gray points) and experimental results (black points). The growth strength calculated from each simulation was interpolated over the range of peak pressures and growth factors seen in experiment for samples with 2D (a) and 3D (b) initial perturbations. Curves of constant growth strength are in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
[5] Tong W. Pressure-shear stress wave analysis in plate impact experiments. Int J Impact Eng 1997;19(2):147–64. [6] Meyers MA, Gregori F, Kad BK, Schneider MS, Kalantar DH, Remington BA, et al. Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Materialia 2003;51(5):1211–28. doi:10.1016/S1359-6454(02)00420-2. [7] Murphy WJ, Higginbotham A, Kimminau G, Barbrel B, Bringa EM, Hawreliak J, et al. The strength of single crystal copper under uniaxial shock compression at 100 GPa. J Phys 2010;22(6):065404. doi:10.1088/0953-8984/22/6/065404. [8] Wallace DC. Nature of the process of overdriven shocks in metals. Phys Rev B 1981;24(10):5607–15. doi:10.1103/PhysRevB.24.5607. [9] Preston DL, Tonks DL, Wallace DC. Model of plastic deformation for extreme loading conditions. J Appl Phys 2003;93(1):211–20. doi:10.1063/1.1524706. [10] Richtmyer RD. Taylor instability in shock acceleration of compressible fluids. Commun Pure Appl Math 1960;13(2):297–319. doi:10.1002/cpa.3160130207. [11] Meshkov EE. Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn 1969;4(5):101–4. doi:10.1007/BF01015969. [12] Prime MB, Vaughan DE, Preston DL, Buttler WT, Chen SR, Oró DM, et al. Using growth and arrest of richtmyer-meshkov instabilities and lagrangian simulations to study high-rate material strength. J Phys 2014;500(11):112051. doi:10.1088/1742-6596/500/11/112051. [13] Prime MB, Buttler WT, Sjue SK, Jensen BJ, Mariam FG, Oró DM, et al. Using Richtmyer-Meshkov instabilities to estimate metal strength at very high rates. In: Song B, Lamberson L, Casem D, Kimberley J, editors. Dynamic Behavior of Materials, Volume 1. Springer International Publishing; 2016. p. 191–7. ISBN:978-3-31922451-0 978-3-319-22452-7, doi: 10.1007/978-3-319-22452-7_27. [14] Prime MB, Buttler WT, Buechler MA, Denissen NA, Kenamond MA, Mariam FG, et al. Estimation of metal strength at very high rates using free-surface richtmyermeshkov instabilities. J Dyn Behav Mater 2017:1–14. doi:10.1007/s40870-017-0103-9. [15] Maddox BR, Park HS, Lu CH, Remington BA, Prisbrey S, Kad B, et al. Isentropic/shock compression and recovery methodology for materials using high-amplitude laser pulses. Mater Sci Eng 2013;578:354–61. doi:10.1016/j.msea.2013.04.050. [16] Randall GC, Vecchio J, Knipping J, Wall D, Remington T, Fitzsimmons P, et al. Developments in microcoining rippled metal foils. Fus Sci Technol 2013;63(2):274–81 URL http://epubs.ans.org/?a=16350. [17] Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Courier Corporation; 2013. [18] Jacobs JW, Catton I. Three-dimensional rayleigh-taylor instability part 2. experiment. J Fluid Mech 1988;187:353–71. doi:10.1017/S0022112088000461. [19] Barker LM, Hollenbach RE. Laser interferometer for measuring high velocities of any reflecting surface. J Appl Phys 1972;43(11):4669–75. doi:10.1063/1.1660986. [20] Celliers PM, Bradley DK, Collins GW, Hicks DG, Boehly TR, Armstrong WJ. Lineimaging velocimeter for shock diagnostics at the OMEGA laser facility. Rev Sci Instrum 2004;75(11):4916–29. doi:10.1063/1.1807008. [21] Larsen JT, Lane SM. HYADES a plasma hydrodynamics code for dense plasma studies. J Quant Spectrosc Radiat Transf 1994;51(12):179–86. doi:10.1016/0022-4073(94)90078-7. [22] McGlaun JM, Thompson SL, Elrick MG. CTH: A three-dimensional shock wave physics code. Int J Impact Eng 1990;10(1):351–60. doi:10.1016/0734-743X(90)90071-3. [23] Holmes NC. Shock compression of low-density foams. In: AIP Conference Proceedings, 309. AIP Publishing; 1994. p. 153–6. doi:10.1063/1.46377. [24] Bennett BI, Johnson JD, Kerley GI, Rood GT. Recent developments in the sesame equation-of-state library. Tech. Rep., LA-7130. Los Alamos Scientific Lab., N.Mex. (USA); 1978. URL http://www.osti.gov/scitech/biblio/5150206. [25] Steinberg DJ, Cochran SG, Guinan MW. A constitutive model for metals applicable at high-strain rate. J Appl Phys 1980;51(3):1498–504. doi:10.1063/1.327799. [26] Steinberg DJ. Equation of state and strength properties of selected materials. Tech. Rep., UCRL-MA-106439 (change 1). Lawrence Livermore National Laboratory, CA (USA); 1996.
Fig. 10. A comparison between the strength inferred from recovered samples with 2D and 3D initial perturbations and the PTW model. Two PTW curves using different states were calculated for comparison. The first curve used 𝑃 = 0 GPa, 𝑇 = 300 K, and plastic strain 𝜀 = 0.25 as a lower bound of growth states. The second curved is an upper bound with 𝑃 = 5 GPa, 𝑇 = 600 K, and 𝜀 = 1.5.
Acknowledgments This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Security, LLC, Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This work was supported by the NNSA through the HEDLP program, grant numbers DE-NA0001805 and DE-NA0001832. Experimental time was awarded by the Jupiter Laser Facility at Lawrence Livermore National Laboratory. We would like to thank the staff at the Jupiter Laser Facility for their help during our experimental campaign. The experiment was completed with the help of Professor Marc Meyers, Shiteng Zhao, and Eric Hahn of the University of California, San Diego; Suzanne Ali of the University of California, Berkeley; and Laura Chen of Imperial College, London. References [1] Kumar A, Kumble RG. Viscous drag on dislocations at high strain rates in copper. J Appl Phys 1969;40(9):3475–80. doi:10.1063/1.1658222. [2] Follansbee PS, Gray GT. Dynamic deformation of shock prestrained copper. Mater Sci Eng 1991;138(1):23–31. doi:10.1016/0921-5093(91)90673-B. [3] Frutschy KJ, Clifton RJ. High-temperature pressure-shear plate impact experiments on OFHC copper. J Mech Phys Solids 1998;46(10):1723–44. doi:10.1016/S0022-5096(98)00055-6. [4] Clifton RJ. Response of materials under dynamic loading. Int J Solids Struct 2000;37(12):105–13. doi:10.1016/S0020-7683(99)00082-7.
6